A308409 -id:A308409 - cp's OEIS frontend

A333080 Number of fixed Tangles of size n.

1, 2, 6, 22, 88, 372, 1626, 7292, 33309, 154374, 723740, 3425124, 16336747, 78437858

Offset: 0

Douglas A. Torrance, Mar 07 2020

a(n) is the number of fixed Tangles (smooth simple closed curves piecewise-defined by quadrants of circles) which have a dual graph containing n edges, or equivalently, enclose an area of (4*n + Pi)*r^2, where 1/r is the curvature.  By 'fixed', we mean that we do not allow rotations or reflections.
Dual graphs of Tangles are polyedges (A096267), but the only chordless cycles allowed are squares, e.g., this is *not* the dual graph of a Tangle:
    o-o-o
    |   |
    o-o-o
  but this is:
    o-o-o
    | | |
    o-o-o

Douglas A. Torrance, Enumeration of planar Tangles, arXiv:1906.01541 [math.CO], 2019-2020. Sums of rows from Table 4.1 (A).

Dual graphs of Tangles which are trees are bond trees on the square lattice (A308409), free Tangles (A333233).

a(11)-a(13) from John Mason, Feb 14 2023

A385120 Number of fixed tree-like polyedges on the square lattice with n edges, rooted at a vertex.

Original entry on oeis.org

1, 4, 18, 88, 435, 2184, 11018, 55888, 284229, 1448800, 7396290, 37804344, 193405121, 990117104, 5072380140

Offset: 0

Ben Samberg, Jun 18 2025

			a(0) = 1: empty structure.
a(1) = 4: a single vertical or horizontal edge, rooted at one of the two vertices.
a(2) = 18: six unrooted two-edge polyedges (a straight path oriented in 2 possible ways and an L-shaped path oriented in 4 possible ways), each rooted at one of the three vertices.

Ben Samberg, a(2) visualization
Ben Samberg, Python algorithm

Cf. A096267 (not necessarily treelike), A056841 (free), A066158 (polyominoes).

Cf. A308409.

a(n) = A308409(n) * (n+1). - Andrei Zabolotskii, Jul 02 2025

cp's OEIS Frontend

A333080 Number of fixed Tangles of size n.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions